SUSY Statistical Mechanics and Random Band Matrices

  • Thomas Spencer
Part of the Lecture Notes in Mathematics book series (LNM, volume 2051)


Ordinary matter is composed of electrons (negatively charged) and nuclei (positively charged) interacting via electromagnetic forces.


Sigma Model Ward Identity Random Environment Grassmann Variable Quantum Diffusion 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



I wish to thank my collaborators Margherita Disertori and Martin Zirnbauer for making this review on supersymmetry possible. Thanks also to Margherita Disertori and Sasha Sodin for numerous helpful comments and corrections on early versions of this review and to Yves Capdeboscq for improving the lower bound in (4.88). I am most grateful to the organizers of the C.I.M.E. summer school, Alessandro Giuliani, Vieri Mastropietro, and Jacob Yngvason for inviting me to give these lectures and for creating a stimulating mathematical environment. Finally, I thank Abdelmalek Abdesselam for many helpful questions during the course of these lectures.


  1. 1.
    A. Abdesselam, Grassmann-Berezin calculus and theorems of the matrix-tree type. Adv. Appl. Math. 33, 51–70 (2004).
  2. 2.
    E. Abrahams, P.W. Anderson, D.C. Licciardello, T.V. Ramakrishnan, Scaling theory of localization: Absence of quantum diffusion in two dimensions. Phys. Rev. Lett. 42, 673 (1979)Google Scholar
  3. 3.
    E.J. Beamond, A.L. Owczarek, J. Cardy, Quantum and classical localisation and the Manhattan lattice. J. Phys. A Math. Gen. 36, 10251 (2003) [arXiv:cond-mat/0210359]Google Scholar
  4. 4.
    F.A. Berezin, Introduction to Superanalysis (Reidel, Dordrecht, 1987)Google Scholar
  5. 5.
    H. Brascamp, E. Lieb, On extensions of the Brunn-Minkowski and Prekopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation. J. Func. Anal. 22, 366–389 (1976)Google Scholar
  6. 6.
    E. Brezin, V. Kazakov, D. Serban, P. Wiegman, A. Zabrodin, in Applications of Random Matrices to Physics. Nato Science Series, vol. 221 (Springer, Berlin, 2006)Google Scholar
  7. 7.
    F. Constantinescu, G. Felder, K. Gawedzki, A. Kupiainen, Analyticity of density of states in a gauge-invariant model for disordered electronic systems. J. Stat. Phys. 48, 365 (1987)Google Scholar
  8. 8.
    P. Deift, Orthogonal Polynomials, and Random Matrices: A Riemann-Hilbert Approach (CIMS, New York University, New York, 1999)Google Scholar
  9. 9.
    P. Deift, T. Kriecherbauer, K.T.-K. McLaughlin, S. Venakides, X. Zhou, Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory. Comm. Pure Appl. Math. 52, 1335–1425 (1999)Google Scholar
  10. 10.
    P. Diaconis, in Recent Progress on de Finetti’s Notions of Exchangeability. Bayesian Statistics, vol. 3 (Oxford University Press, New York, 1988), pp. 111–125Google Scholar
  11. 11.
    M. Disertori, Density of states for GUE through supersymmetric approach. Rev. Math. Phys. 16, 1191–1225 (2004)Google Scholar
  12. 12.
    M. Disertori, T. Spencer, Anderson localization for a SUSY sigma model. Comm. Math. Phys. 300, 659–671 (2010).
  13. 13.
    M. Disertori, H. Pinson T. Spencer, Density of states for random band matrices. Comm. Math. Phys. 232, 83–124 (2002). Google Scholar
  14. 14.
    M. Disertori, T. Spencer, M.R. Zirnbauer, Quasi-diffusion in a 3D supersymmetric hyperbolic sigma model. Comm. Math Phys. 300, 435 (2010).
  15. 15.
    K.B. Efetov, in Anderson Localization and Supersymmetry, ed. by E. Abrahams. 50 Years of Anderson Localization (World Scientific, Singapore, 2010).
  16. 16.
    K.B. Efetov, Minimum metalic conductivity in the theory of localization. JETP Lett. 40, 738 (1984)Google Scholar
  17. 17.
    K.B. Efetov, Supersymmetry and theory of disordered metals. Adv. Phys. 32, 874 (1983)Google Scholar
  18. 18.
    K.B. Efetov, Supersymmetry in Disorder and Chaos (Cambridge University Press, Cambridge, 1997)Google Scholar
  19. 19.
    L. Erdős, Universality of Wigner random matrices: A survey of recent results. Russian Math. Surveys. 66(3) 507–626 (2011).
  20. 20.
    L. Erdős, B. Schlein, H-T. Yau, Local semicircle law and complete delocalization for Wigner random matrices. Comm. Math. Phys. 287, 641–655 (2009)Google Scholar
  21. 21.
    L. Erdős, M. Salmhofer, H-T. Yau, Quantum diffusion of the random Schrödinger evolution in the scaling limit. Acta Math. 200, 211 (2008)Google Scholar
  22. 22.
    L. Erdős, in Les Houches Lectures: Quantum Theory from Small to Large Scales, Lecture notes on quantum Brownian motion, vol. 95,
  23. 23.
    L. Erdős, H.T. Yau, J. Yin, Bulk universality for generalized Wigner matrices. to appear in Prob. Th. and Rel.Fields. [arxiv:1001.3453]Google Scholar
  24. 24.
    J. Feldman, H. Knörrer, E. Trubowitz, in Fermionic Functional Integrals and the Renormalization Group. CRM Monograph Series (AMS, Providence, 2002)Google Scholar
  25. 25.
    J. Fröhlich, T. Spencer, The Kosterlitz Thouless transition. Comm. Math. Phys. 81, 527 (1981)Google Scholar
  26. 26.
    J. Fröhlich, T. Spencer, On the statistical mechanics of classical Coulomb and dipole gases. J. Stat. Phys. 24, 617–701 (1981)Google Scholar
  27. 27.
    J. Fröhlich, B. Simon, T. Spencer, Infrared bounds, phase transitions and continuous symmetry breaking. Comm. Math. Phys. 50, 79 (1976)Google Scholar
  28. 28.
    Y.V. Fyodorov, in Basic Features of Efetov’s SUSY , ed. by E. Akkermans et al. Mesoscopic Quantum Physics (Les Houches, France, 1994)Google Scholar
  29. 29.
    Y.V. Fyodorov, Negative moments of characteristic polynomials of random matrices. Nucl. Phys. B 621, 643–674 (2002).
  30. 30.
    I.A. Gruzberg, A.W.W. Ludwig, N. Read, Exact exponents for the spin quantun Hall transition. Phys. Rev. Lett. 82, 4524–4527 (1999)Google Scholar
  31. 31.
    T. Guhr, in Supersymmetry in Random Matrix Theory. The Oxford Handbook of Random Matrix Theory (Oxford University Press, Oxford, 2010). http://arXiv:1005.0979v1Google Scholar
  32. 32.
    J.M. Kosterlitz, D.J. Thouless, Ordering, metastability and phase transitions in two-dimensional systems. J. Phys. C 6, 1181 (1973)Google Scholar
  33. 33.
    P. Littelmann, H.-J. Sommers, M.R. Zirnbauer, Superbosonization of invariant random matrix ensembles. Comm. Math. Phys. 283, 343–395 (2008)Google Scholar
  34. 34.
    F. Merkl, S. Rolles, in Linearly Edge-Reinforced Random Walks. Dynamics and Stochastics. Lecture Notes-Monograph Series, vol. 48 (2006), pp. 66–77Google Scholar
  35. 35.
    F. Merkl, S. Rolles, Edge-reinforced random walk on one-dimensional periodic graphs. Probab. Theor. Relat. Field 145, 323 (2009)Google Scholar
  36. 36.
    A.D. Mirlin, in Statistics of Energy Levels, ed. by G.Casati, I. Guarneri, U. Smilansky. New Directions in Quantum Chaos. Proceedings of the International School of Physics “Enrico Fermi”, Course CXLIII (IOS Press, Amsterdam, 2000), pp. 223–298.
  37. 37.
    L. Pastur, M. Shcherbina, Universality of the local eigenvalue statistics for a class of unitary invariant matrix ensembles. J. Stat. Phys. 86, 109–147 (1997)Google Scholar
  38. 38.
    R. Peamantle, Phase transition in reinforced random walk and RWRE on trees. Ann. Probab. 16, 1229–1241 (1988)Google Scholar
  39. 39.
    Z. Rudnick, P. Sarnak, Zeros of principal L-functions and random-matrix theory. Duke Math. J. 81, 269–322 (1996)Google Scholar
  40. 40.
    M. Salmhofer, Renormalization: An Introduction (Springer, Berlin, 1999)Google Scholar
  41. 41.
    L. Schäfer, F. Wegner, Disordered system with n orbitals per site: Lagrange formulation, hyperbolic symmetry, and Goldstone modes. Z. Phys. B 38, 113–126 (1980)Google Scholar
  42. 42.
    J. Schenker, Eigenvector localization for random band matrices with power law band width. Comm. Math. Phys. 290, 1065–1097 (2009)Google Scholar
  43. 43.
    A. Sodin, The spectral edge of some random band matrices. Ann. Math. 172, 2223 (2010)Google Scholar
  44. 44.
    A. Sodin, An estimate on the average spectral measure for random band matrices, J. Stat. Phys. 144, 46–59 (2011). Google Scholar
  45. 45.
    T. Spencer, in Mathematical Aspects of Anderson Localization, ed. by E. Abrahams. 50 Years of Anderson Localization (World Scientific, Singapore, 2010)Google Scholar
  46. 46.
    T. Spencer, in Random Band and Sparse Matrices. The Oxford Handbook of Random Matrix Theory, eds. by G. Akermann, J. Baik, and P. Di Francesco. (Oxford University Press, Oxford, 2010)Google Scholar
  47. 47.
    T. Spencer, M.R. Zirnbauer, Spontaneous symmetry breaking of a hyperbolic sigma model in three dimensions. Comm. Math. Phys. 252, 167–187 (2004). Google Scholar
  48. 48.
    B. Valko, B. Virag, Random Schrödinger operators on long boxes, noise explosion and the GOE,
  49. 49.
    J.J.M. Verbaarschot, H.A. Weidenmüller, M.R. Zirnbauer, Phys. Rep. 129, 367–438 (1985)Google Scholar
  50. 50.
    F. Wegner, The Mobility edge problem: Continuous symmetry and a Conjecture. Z. Phys. B 35, 207–210 (1979)Google Scholar
  51. 51.
    M.R. Zirnbauer, The Supersymmetry method of random matrix theory,
  52. 52.
    M.R. Zirnbauer, Fourier analysis on a hyperbolic supermanifold with constant curvature. Comm. Math. Phys. 141, 503–522 (1991)Google Scholar
  53. 53.
    M.R. Zirnbauer, Riemannian symmetric superspaces and their origin in random-matrix theory. J. Math. Phys. 37, 4986–5018 (1996)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.School of MathematicsInstitute for Advanced StudyPrincetonUSA

Personalised recommendations